Analysis of the Rational Krylov Subspace and ADI Methods for Solving the Lyapunov Equation

Druskin, Vladimir; Knizhnerman, Leonid; Simoncini, Valeria

doi:10.1137/100813257

Cited by 98 publications

(135 citation statements)

References 40 publications

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“…For effective methods to solve large-scale Lyapunov equations, see, e.g., [46,47,152,219] for ADI-type methods, [117,191,199,206] for the Smith method and its variants, [78,135,203,210] for Krylov-based methods, or the recent survey [48] and the references therein.…”

Section: Balanced Truncationmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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Abstract. Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification-settings that require repeated model evaluations over different parameter values.

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Section: Balanced Truncationmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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“…. , k. These factors can efficiently be computed by the lowrank alternating directions implicit (LR-ADI) method [5,19,24], the rational Krylov method [9,26] or the Riemannian method [30]. Then for a reduced basis matrix V k = [ Z 1 , .…”

Section: Offline Phasementioning

confidence: 99%

“…To avoid expensive computations in the direct methods, iterative ones such as the sign function method [6], the alternating directions implicit method [20,33], and Krylov subspace methods [16,25] have been developed. Exploiting the fact that the right-hand sides of Lyapunov equations in most applications have low rank, low-rank versions of the mentioned iterative methods have been formulated, [9,19,24,26], just to name a few, see also [5,27] for the recent surveys on the state-of-the-art algorithms.…”

Section: Introduction In This Paper We Consider the Following Parammentioning

confidence: 99%

Solving Parameter-Dependent Lyapunov Equations Using the Reduced Basis Method with Application to Parametric Model Order Reduction

Sơn¹,

Stykel²

2017

SIAM J. Matrix Anal. & Appl.

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Abstract. Our aim is to numerically solve parameter-dependent Lyapunov equations using reduced basis method. Such equations arise in parametric model order reduction. We restrict ourselves to systems that affinely depend on the parameter as our main ingredient is the min-Θ approach. In those cases, we derive various a posteriori error estimates. Based on these estimates, Greedy algorithms for constructing reduced basis are formulated. Thanks to the derived results, a novel so-called parametric balanced truncation model reduction method is developed. Numerical examples are presented.

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“…The convergence of the Galerkin method on tensor products of (rational) Krylov subspaces for Lyapunov and Sylvester equations has been analysed in [2,10,21,20,27,28]. For the extensions considered in this paper the framework developed in [2] appears to be most suitable.…”

Section: Introductionmentioning

confidence: 99%

An Error Analysis of Galerkin Projection Methods for Linear Systems with Tensor Product Structure

Beckermann

Kreßner²,

Tobler³

2013

SIAM J. Numer. Anal.

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Recent results on the convergence of a Galerkin projection method for the Sylvester equation are extended to more general linear systems with tensor product structure. In the Hermitian positive definite case, explicit convergence bounds are derived for Galerkin projection based on tensor products of rational Krylov subspaces. The results can be used to optimize the choice of shifts for these methods. Numerical experiments demonstrate that the convergence rates predicted by our bounds appear to be tight.Linear system, Kronecker product structure, Sylvester equation, tensor projection, Galerkin projection, rational Krylov subspaces.15A24, 65F10.

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Analysis of the Rational Krylov Subspace and ADI Methods for Solving the Lyapunov Equation

Cited by 98 publications

References 40 publications

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

Solving Parameter-Dependent Lyapunov Equations Using the Reduced Basis Method with Application to Parametric Model Order Reduction

An Error Analysis of Galerkin Projection Methods for Linear Systems with Tensor Product Structure

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